Optimal. Leaf size=125 \[ -\frac{\left (a+\frac{1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{24 \left (a+\frac{1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{46 \left (a+\frac{1}{x}\right )}{35 a^2 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{35 a+\frac{13}{x}}{35 a^2 c^6 \sqrt{1-\frac{1}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.410555, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6175, 6178, 852, 1635, 637} \[ -\frac{\left (a+\frac{1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{24 \left (a+\frac{1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{46 \left (a+\frac{1}{x}\right )}{35 a^2 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{35 a+\frac{13}{x}}{35 a^2 c^6 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6175
Rule 6178
Rule 852
Rule 1635
Rule 637
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx &=\frac{\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^6 x^6} \, dx}{a^6 c^6}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1-\frac{x}{a}\right )^3 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a^6 c^6}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (1+\frac{x}{a}\right )^3}{\left (1-\frac{x^2}{a^2}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{a^6 c^6}\\ &=-\frac{\left (a+\frac{1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^2 \left (3 a^4+7 a^3 x+7 a^2 x^2+7 a x^3\right )}{\left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 a^6 c^6}\\ &=\frac{24 \left (a+\frac{1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{\left (a+\frac{1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right ) \left (33 a^4+70 a^3 x+35 a^2 x^2\right )}{\left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{35 a^6 c^6}\\ &=-\frac{46 \left (a+\frac{1}{x}\right )}{35 a^2 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{24 \left (a+\frac{1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{\left (a+\frac{1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{\operatorname{Subst}\left (\int \frac{39 a^4+105 a^3 x}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{105 a^6 c^6}\\ &=-\frac{46 \left (a+\frac{1}{x}\right )}{35 a^2 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{24 \left (a+\frac{1}{x}\right )^2}{35 a^3 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{\left (a+\frac{1}{x}\right )^3}{7 a^4 c^6 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{35 a+\frac{13}{x}}{35 a^2 c^6 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0705729, size = 66, normalized size = 0.53 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13\right )}{35 c^6 (a x-1)^4 (a x+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 66, normalized size = 0.5 \begin{align*}{\frac{ \left ( 8\,{x}^{4}{a}^{4}-24\,{x}^{3}{a}^{3}+20\,{a}^{2}{x}^{2}+4\,ax-13 \right ) \left ( ax+1 \right ) }{35\, \left ( ax-1 \right ) ^{5}{c}^{6}a} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05046, size = 131, normalized size = 1.05 \begin{align*} \frac{1}{560} \, a{\left (\frac{35 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{6}} + \frac{\frac{28 \,{\left (a x - 1\right )}}{a x + 1} - \frac{70 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{140 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{6} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5806, size = 204, normalized size = 1.63 \begin{align*} \frac{{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{5} c^{6} x^{4} - 4 \, a^{4} c^{6} x^{3} + 6 \, a^{3} c^{6} x^{2} - 4 \, a^{2} c^{6} x + a c^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (a c x - c\right )}^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]